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%I #38 Apr 22 2024 07:35:53
%S 1,4,72,3456,345600,62208000,18289152000,8193540096000,
%T 5309413982208000,4778472583987200000,5781951826624512000000,
%U 9158611693373227008000000,18573664514160904372224000000,47325697182081984340426752000000,149075946123558250672344268800000000
%N a(n) = n!*((n+1)!)^2.
%C 2*a(n-1) is the number of elements of the wreath product of S_n and S_3 with cycle partition equal to (3n). - _Josaphat Baolahy_, Mar 12 2024
%F G.f. of hypergeometric type: Sum_{n>=0} a(n)*z^n/(n!)^3 = (1+z)/(1-z)^3.
%F Integral representation as n-th moment of a positive function w(x) on a positive half axis (solution of the Stieltjes moment problem), in Maple notation: a(n) = int(x^n*w(x),x=0..infinity), n>=0, where w(x) = MeijerG([[],[]],[[1,1,0]],[]],x), w(0)=1, limit(w(x),x=infinity)=0.
%F w(x) is monotonically decreasing over (0,infinity).
%F The Meijer G function above cannot be represented by any other known special function.
%F This solution of the Stieltjes moment problem is not unique.
%F Asymptotics: a(n) -> (1/960)*sqrt(2)*Pi^(3/2)*(1920*n^3 + 4320*n^2 + 2940*n + 589)*exp(-3*n)*n^(1/2 + 3*n), for n->oo.
%F a(n) = A172492(n)*(n+1).
%F a(n) - n*(n+1)^2*a(n-1) = 0. - _R. J. Mathar_, Jul 28 2013
%t Table[n!*((n+1)!)^2,{n,0,15}]
%o (Python)
%o from math import factorial
%o def A224900(n): return factorial(n)**3*(n+1)**2 # _Chai Wah Wu_, Apr 22 2024
%Y Cf. A172492.
%K nonn,easy
%O 0,2
%A _Karol A. Penson_, Jul 24 2013
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